Question

Let k be the ratio of the supplement of an acute angle to that of its complement. Then, k always satisfy which of the following conditions?

• A. $0<k<\frac{1}{2}$
  
• B. $\frac{1}{2}<k<1$
  
• C. 1 < k < 2
• D. k > 2

Answer 1

The correct option is D k > 2

Let the acute angle be x.
Thus, the supplement of x is (180° – x) and the complement of x is (90° – x).
Since k is the ratio of supplement of an angle to its complement.
$\therefore k=\frac{{180}^{\circ }-x}{{90}^{\circ }-x}$
$=\frac{{180}^{\circ }-x+x-x}{{90}^{\circ }-x}$
$=\frac{{180}^{\circ }-2x+x}{{90}^{\circ }-x}$
$=\frac{2({90}^{\circ }-x)+x}{{90}^{\circ }-x}$
$=2+\frac{x}{{90}^{\circ }-x}$
$\Rightarrow k>2$ $(\because 0^{\circ }<x<{90}^{\circ })$
Hence, the correct answer is option (d).